Finite Group DatabasebetaFinite Group DatabasebetaOrder 1344.
560 conjugacy classes of subgroups; only the 273 normal subgroups are shown.
This group has 560 conjugacy classes of subgroups — too many to draw. Only its 273 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
Splittings G = N ⋊ H with N normal; H ≅ G/N.
| N | H ≅ G/N | type |
|---|---|---|
| 1. C22×Q8×ΓL(1,𝔽8) | C2 | semidirect |
| 2. ΓL(1,𝔽8)×C4⋊Q8 | C2 | semidirect |
| 3. ΓL(1,𝔽8)×C22⋊Q8 | C2 | semidirect |
| 4. C4×Q8×ΓL(1,𝔽8) | C2 | semidirect |
| 6. C7×D4×Q8 | C3 | semidirect |
| 17. C22×C7×Q8 | C2×C3 | semidirect |
| 18. C7×C4⋊Q8 | C2×C3 | semidirect |
| 19. C7×C22⋊Q8 | C2×C3 | semidirect |
| 20. C4×C7×Q8 | C2×C3 | semidirect |
| 25. Q8×ΓL(1,𝔽8) | D4 | direct (N × H) |
| 26. D4×ΓL(1,𝔽8) | Q8 | direct (N × H) |
| 49. C22×ΓL(1,𝔽8) | C2×Q8 | semidirect |
| 54. C4×ΓL(1,𝔽8) | C2×Q8 | semidirect |
| 55. D4×Q8 | ΓL(1,𝔽8) | direct (N × H) |
| 58. C7×Q8 | C3×D4 | semidirect |
| 59. C7×D4 | C3×Q8 | semidirect |
| 79. C22×Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 80. C4⋊Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 81. C22⋊Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 82. C4×Q8 | C2×ΓL(1,𝔽8) | semidirect |
| 86. C22×C7 | C2×C3×Q8 | semidirect |
| 91. C4×C7 | C2×C3×Q8 | semidirect |
| 101. ΓL(1,𝔽8) | D4×Q8 | direct (N × H) |
| 126. Q8 | D4×ΓL(1,𝔽8) | direct (N × H) |
| 127. D4 | Q8×ΓL(1,𝔽8) | direct (N × H) |
| 133. C7 | C3×D4×Q8 | semidirect |
| 140. C22 | C2×Q8×ΓL(1,𝔽8) | semidirect |
| 145. C4 | C2×Q8×ΓL(1,𝔽8) | semidirect |
By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.
| subgroup | index |
|---|---|
| C22×Q8×ΓL(1,𝔽8) | 2 |
| ΓL(1,𝔽8)×C4⋊Q8 | 2 |
| ΓL(1,𝔽8)×C22⋊Q8 | 2 |
| C4×Q8×ΓL(1,𝔽8) | 2 |
| C4×D4×ΓL(1,𝔽8) | 2 |
| C7×D4×Q8 | 3 |
| C3×D4×Q8 | 7 |